## Elliptic Integral of the 1st Kind

This tool evaluates the complete or incomplete elliptic integral of the first kind: K(k) or F(φ,k) respectively. Select the desired type of the calculation and enter the appropriate arguments below.

 Type Complete Incomplete k = Result:

## About Elliptic Integrals of the First Kind

### Definitions

The incomplete elliptic integral of the first kind is defined as:

where k is the elliptic modulus, with . Variable is the Jacobi's amplitude.

The complete elliptic integral of the first kind is defined as:

### Values

In the following table the values of the complete elliptic integral of the first kind are shown for a range of k values:

In the following table the values of the incomplete elliptic integral of the first kind are shown for a range of k and φ values:

The xy plots of the incomplete integral of the first kind for various values of amplitude φ are depicted in the following figure: